Abstract

The method of superimposing multiple phase patterns to generate and deflect multi Airy beams is proposed in this paper. A Dammann grating and an optimized splitting grating are superimposed, respectively, with an Airy cubic phase pattern to generate an array of 4×4 equal-space Airy beams. By adding a deflection grating to the superimposed phase patterns, the transverse self-accelerated Airy beams array can be deflected arbitrarily in two-dimensional plane. The impacts of superimposed phase patterns on the transverse acceleration and size of main lobe of Airy beams in array are discussed in this paper. Meanwhile, the accuracy of the steering method and the impact of the phase modulation depth on the size of the Airy beams are introduced.

© 2013 Optical Society of America

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  1. G. A. Siviloglou and D. N. Christodoulides, “Observation of accelerating Airy beams,” Opt. Lett. 32, 979–981 (2007).
    [CrossRef]
  2. T. Ellenbogen, N. Voloch-Bloch, A. Ganany-Padowicz, and A. Aire, “Nonlinear generation and manipulation of Airy beams,” Nat. Photonics 3, 395–398 (2009).
    [CrossRef]
  3. J. Baumgartl, M. Mzailu, and K. Dholakia, “Optically mediated particle clearing using Airy wavepackets,” Nat. Photonics 2, 675–678 (2008).
    [CrossRef]
  4. R. D. Leonardo, F. Ianni, and G. Rucco, “Computer generation of optimal holograms for optical trap arrays,” Opt. Express 15, 1913–1922 (2007).
    [CrossRef]
  5. D. Palima and V. R. Daria, “Holographic projection of arbitrary light patterns with a suppressed zero-order beam,” Appl. Opt. 46, 4197–4201 (2007).
    [CrossRef]
  6. P. Garcia-Martinez, M. M. Sanchez-Lopez, J. A. Davis, D. M. Cottrell, D. Sand, and I. Moreno, “Generation of Bessel beam arrays through Dammann gratings,” Appl. Opt. 51, 1375–1381 (2012).
    [CrossRef]
  7. J. Yu, C. Zhou, W. Jia, A. Hu, W. Cao, J. Wu, and S. Wang, “Three-dimensional Dammann vortex array with tunable topological charge,” Appl. Opt. 51, 2485–2490 (2012).
    [CrossRef]
  8. Y. Shinoda, J. P. Liu, P. S. Chung, K. Dobson, X. Zhou, and T. C. Poon, “Three-dimensional complex image coding using a circular Dammann grating,” Appl. Opt. 50, B38–B45 (2011).
    [CrossRef]
  9. J. A. Davis, I. Moreno, J. L. Martinex, T. J. Hernandez, and D. M. Cottrell, “Creating three-dimensional lattice patterns using programmable Dammann gratings,” Appl. Opt. 50, 3653–3657 (2011).
    [CrossRef]
  10. L. A. Romero, and F. M. Dickey, “Theory of optimal beam splitting by phase gratings. I. One-dimensional gratings,” J. Opt. Soc. Am. A 24, 2280–2295 (2007).
    [CrossRef]
  11. J. Albero, and I. Moreno, “Grating beam splitting with liquid crystal adaptive optics,” J. Opt. 14, 075704 (2012).
    [CrossRef]
  12. J. Albero, I. Moreno, J. A. Davis, D. M. Cottrell, and D. Sand, “Generalized phase diffraction gratings with tailored intensity,” Opt. Lett. 37, 4227–4229 (2012).
    [CrossRef]
  13. Y. Hu, P. Zhang, C. Lou, S. Huang, J. Xu, and Z. Chen, “Optimal control of the ballistic motion of Airy beams,” Opt. Lett. 35, 2260–2262 (2010).
    [CrossRef]
  14. Y. Hu, G. A. Siviloglou, P. Zhang, N. K. Efremidis, D. N. Christodoulides, and Z. Chen, “Self-accelerating Airy beams: generation, control, and applications,” in Nonlinear Photonics and Novel Optical Phenomena, Z. Chen and M. Roberto, eds., Springer Series in Optical Sciences (Springer, 2012), Vol. 171, pp. 1–46.
  15. X.-Z. Wang, Q. Li, and Q. Wang, “Arbitrary scanning of the Airy beams using additional phase grating with cubic phase mask,” Appl. Opt. 51, 6726–6731 (2012).
    [CrossRef]
  16. N. Zhang, X. C. Yuan, and R. E. Burge, “Extending the detection range of optical vortices by Dammann vortex gratings,” Opt. Lett. 35, 3495–3497 (2010).
    [CrossRef]
  17. C. Zhou, and L. Liu, “Numerical study of Dammann array illuminators,” Appl. Opt. 34, 5961–5969 (1995).
    [CrossRef]
  18. F. Gori, M. Santarsiero, S. Vicalvi, R. Borghi, G. Cincotti, E. Di Fabrizio, and M. Gentili, “Analytical derivation of the optimum triplicator,” Opt. Commun. 157, 13–16(1998).
    [CrossRef]
  19. D. Engstrom, J. Bengtsson, E. Eriksson, and M. Goksor, “Improved beam steering accuracy of a single beam with a 1D phase-only spatial light modulator,” Opt. Express 16, 18275–18287 (2008).
    [CrossRef]
  20. P. F. McManamon, “A review of phased array steering for narrow-band electrooptical systems,” Proc. IEEE 97, 1078–1096 (2009).
    [CrossRef]
  21. A. Lizana, A. Marquez, L. Lobato, Y. Rodange, I. Moreno, C. Iemmi, and J. Campos, “The minimum Euclidean distance principle applied to improve the modulation diffraction efficiency in digitally controlled spatial light modulators,” Opt. Express 18, 10581–10593 (2010).
    [CrossRef]

2012 (5)

2011 (2)

2010 (3)

2009 (2)

P. F. McManamon, “A review of phased array steering for narrow-band electrooptical systems,” Proc. IEEE 97, 1078–1096 (2009).
[CrossRef]

T. Ellenbogen, N. Voloch-Bloch, A. Ganany-Padowicz, and A. Aire, “Nonlinear generation and manipulation of Airy beams,” Nat. Photonics 3, 395–398 (2009).
[CrossRef]

2008 (2)

J. Baumgartl, M. Mzailu, and K. Dholakia, “Optically mediated particle clearing using Airy wavepackets,” Nat. Photonics 2, 675–678 (2008).
[CrossRef]

D. Engstrom, J. Bengtsson, E. Eriksson, and M. Goksor, “Improved beam steering accuracy of a single beam with a 1D phase-only spatial light modulator,” Opt. Express 16, 18275–18287 (2008).
[CrossRef]

2007 (4)

1998 (1)

F. Gori, M. Santarsiero, S. Vicalvi, R. Borghi, G. Cincotti, E. Di Fabrizio, and M. Gentili, “Analytical derivation of the optimum triplicator,” Opt. Commun. 157, 13–16(1998).
[CrossRef]

1995 (1)

Aire, A.

T. Ellenbogen, N. Voloch-Bloch, A. Ganany-Padowicz, and A. Aire, “Nonlinear generation and manipulation of Airy beams,” Nat. Photonics 3, 395–398 (2009).
[CrossRef]

Albero, J.

Baumgartl, J.

J. Baumgartl, M. Mzailu, and K. Dholakia, “Optically mediated particle clearing using Airy wavepackets,” Nat. Photonics 2, 675–678 (2008).
[CrossRef]

Bengtsson, J.

Borghi, R.

F. Gori, M. Santarsiero, S. Vicalvi, R. Borghi, G. Cincotti, E. Di Fabrizio, and M. Gentili, “Analytical derivation of the optimum triplicator,” Opt. Commun. 157, 13–16(1998).
[CrossRef]

Burge, R. E.

Campos, J.

Cao, W.

Chen, Z.

Y. Hu, P. Zhang, C. Lou, S. Huang, J. Xu, and Z. Chen, “Optimal control of the ballistic motion of Airy beams,” Opt. Lett. 35, 2260–2262 (2010).
[CrossRef]

Y. Hu, G. A. Siviloglou, P. Zhang, N. K. Efremidis, D. N. Christodoulides, and Z. Chen, “Self-accelerating Airy beams: generation, control, and applications,” in Nonlinear Photonics and Novel Optical Phenomena, Z. Chen and M. Roberto, eds., Springer Series in Optical Sciences (Springer, 2012), Vol. 171, pp. 1–46.

Christodoulides, D. N.

G. A. Siviloglou and D. N. Christodoulides, “Observation of accelerating Airy beams,” Opt. Lett. 32, 979–981 (2007).
[CrossRef]

Y. Hu, G. A. Siviloglou, P. Zhang, N. K. Efremidis, D. N. Christodoulides, and Z. Chen, “Self-accelerating Airy beams: generation, control, and applications,” in Nonlinear Photonics and Novel Optical Phenomena, Z. Chen and M. Roberto, eds., Springer Series in Optical Sciences (Springer, 2012), Vol. 171, pp. 1–46.

Chung, P. S.

Cincotti, G.

F. Gori, M. Santarsiero, S. Vicalvi, R. Borghi, G. Cincotti, E. Di Fabrizio, and M. Gentili, “Analytical derivation of the optimum triplicator,” Opt. Commun. 157, 13–16(1998).
[CrossRef]

Cottrell, D. M.

Daria, V. R.

Davis, J. A.

Dholakia, K.

J. Baumgartl, M. Mzailu, and K. Dholakia, “Optically mediated particle clearing using Airy wavepackets,” Nat. Photonics 2, 675–678 (2008).
[CrossRef]

Di Fabrizio, E.

F. Gori, M. Santarsiero, S. Vicalvi, R. Borghi, G. Cincotti, E. Di Fabrizio, and M. Gentili, “Analytical derivation of the optimum triplicator,” Opt. Commun. 157, 13–16(1998).
[CrossRef]

Dickey, F. M.

Dobson, K.

Efremidis, N. K.

Y. Hu, G. A. Siviloglou, P. Zhang, N. K. Efremidis, D. N. Christodoulides, and Z. Chen, “Self-accelerating Airy beams: generation, control, and applications,” in Nonlinear Photonics and Novel Optical Phenomena, Z. Chen and M. Roberto, eds., Springer Series in Optical Sciences (Springer, 2012), Vol. 171, pp. 1–46.

Ellenbogen, T.

T. Ellenbogen, N. Voloch-Bloch, A. Ganany-Padowicz, and A. Aire, “Nonlinear generation and manipulation of Airy beams,” Nat. Photonics 3, 395–398 (2009).
[CrossRef]

Engstrom, D.

Eriksson, E.

Ganany-Padowicz, A.

T. Ellenbogen, N. Voloch-Bloch, A. Ganany-Padowicz, and A. Aire, “Nonlinear generation and manipulation of Airy beams,” Nat. Photonics 3, 395–398 (2009).
[CrossRef]

Garcia-Martinez, P.

Gentili, M.

F. Gori, M. Santarsiero, S. Vicalvi, R. Borghi, G. Cincotti, E. Di Fabrizio, and M. Gentili, “Analytical derivation of the optimum triplicator,” Opt. Commun. 157, 13–16(1998).
[CrossRef]

Goksor, M.

Gori, F.

F. Gori, M. Santarsiero, S. Vicalvi, R. Borghi, G. Cincotti, E. Di Fabrizio, and M. Gentili, “Analytical derivation of the optimum triplicator,” Opt. Commun. 157, 13–16(1998).
[CrossRef]

Hernandez, T. J.

Hu, A.

Hu, Y.

Y. Hu, P. Zhang, C. Lou, S. Huang, J. Xu, and Z. Chen, “Optimal control of the ballistic motion of Airy beams,” Opt. Lett. 35, 2260–2262 (2010).
[CrossRef]

Y. Hu, G. A. Siviloglou, P. Zhang, N. K. Efremidis, D. N. Christodoulides, and Z. Chen, “Self-accelerating Airy beams: generation, control, and applications,” in Nonlinear Photonics and Novel Optical Phenomena, Z. Chen and M. Roberto, eds., Springer Series in Optical Sciences (Springer, 2012), Vol. 171, pp. 1–46.

Huang, S.

Ianni, F.

Iemmi, C.

Jia, W.

Leonardo, R. D.

Li, Q.

Liu, J. P.

Liu, L.

Lizana, A.

Lobato, L.

Lou, C.

Marquez, A.

Martinex, J. L.

McManamon, P. F.

P. F. McManamon, “A review of phased array steering for narrow-band electrooptical systems,” Proc. IEEE 97, 1078–1096 (2009).
[CrossRef]

Moreno, I.

Mzailu, M.

J. Baumgartl, M. Mzailu, and K. Dholakia, “Optically mediated particle clearing using Airy wavepackets,” Nat. Photonics 2, 675–678 (2008).
[CrossRef]

Palima, D.

Poon, T. C.

Rodange, Y.

Romero, L. A.

Rucco, G.

Sanchez-Lopez, M. M.

Sand, D.

Santarsiero, M.

F. Gori, M. Santarsiero, S. Vicalvi, R. Borghi, G. Cincotti, E. Di Fabrizio, and M. Gentili, “Analytical derivation of the optimum triplicator,” Opt. Commun. 157, 13–16(1998).
[CrossRef]

Shinoda, Y.

Siviloglou, G. A.

G. A. Siviloglou and D. N. Christodoulides, “Observation of accelerating Airy beams,” Opt. Lett. 32, 979–981 (2007).
[CrossRef]

Y. Hu, G. A. Siviloglou, P. Zhang, N. K. Efremidis, D. N. Christodoulides, and Z. Chen, “Self-accelerating Airy beams: generation, control, and applications,” in Nonlinear Photonics and Novel Optical Phenomena, Z. Chen and M. Roberto, eds., Springer Series in Optical Sciences (Springer, 2012), Vol. 171, pp. 1–46.

Vicalvi, S.

F. Gori, M. Santarsiero, S. Vicalvi, R. Borghi, G. Cincotti, E. Di Fabrizio, and M. Gentili, “Analytical derivation of the optimum triplicator,” Opt. Commun. 157, 13–16(1998).
[CrossRef]

Voloch-Bloch, N.

T. Ellenbogen, N. Voloch-Bloch, A. Ganany-Padowicz, and A. Aire, “Nonlinear generation and manipulation of Airy beams,” Nat. Photonics 3, 395–398 (2009).
[CrossRef]

Wang, Q.

Wang, S.

Wang, X.-Z.

Wu, J.

Xu, J.

Yu, J.

Yuan, X. C.

Zhang, N.

Zhang, P.

Y. Hu, P. Zhang, C. Lou, S. Huang, J. Xu, and Z. Chen, “Optimal control of the ballistic motion of Airy beams,” Opt. Lett. 35, 2260–2262 (2010).
[CrossRef]

Y. Hu, G. A. Siviloglou, P. Zhang, N. K. Efremidis, D. N. Christodoulides, and Z. Chen, “Self-accelerating Airy beams: generation, control, and applications,” in Nonlinear Photonics and Novel Optical Phenomena, Z. Chen and M. Roberto, eds., Springer Series in Optical Sciences (Springer, 2012), Vol. 171, pp. 1–46.

Zhou, C.

Zhou, X.

Appl. Opt. (7)

J. Opt. (1)

J. Albero, and I. Moreno, “Grating beam splitting with liquid crystal adaptive optics,” J. Opt. 14, 075704 (2012).
[CrossRef]

J. Opt. Soc. Am. A (1)

Nat. Photonics (2)

T. Ellenbogen, N. Voloch-Bloch, A. Ganany-Padowicz, and A. Aire, “Nonlinear generation and manipulation of Airy beams,” Nat. Photonics 3, 395–398 (2009).
[CrossRef]

J. Baumgartl, M. Mzailu, and K. Dholakia, “Optically mediated particle clearing using Airy wavepackets,” Nat. Photonics 2, 675–678 (2008).
[CrossRef]

Opt. Commun. (1)

F. Gori, M. Santarsiero, S. Vicalvi, R. Borghi, G. Cincotti, E. Di Fabrizio, and M. Gentili, “Analytical derivation of the optimum triplicator,” Opt. Commun. 157, 13–16(1998).
[CrossRef]

Opt. Express (3)

Opt. Lett. (4)

Proc. IEEE (1)

P. F. McManamon, “A review of phased array steering for narrow-band electrooptical systems,” Proc. IEEE 97, 1078–1096 (2009).
[CrossRef]

Other (1)

Y. Hu, G. A. Siviloglou, P. Zhang, N. K. Efremidis, D. N. Christodoulides, and Z. Chen, “Self-accelerating Airy beams: generation, control, and applications,” in Nonlinear Photonics and Novel Optical Phenomena, Z. Chen and M. Roberto, eds., Springer Series in Optical Sciences (Springer, 2012), Vol. 171, pp. 1–46.

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Figures (15)

Fig. 1.
Fig. 1.

Dammann grating: (a) normalized coordinates structure diagram of one period, (b) a Dammann grating phase pattern of 512×512 resolution.

Fig. 2.
Fig. 2.

Optimal splitting grating: (a) continuous phase profile in one period, (b) discrete phase gray of the continuous profile in one period, (c) extended phase pattern of 512×512 for 1×4 splitting, (d) extended phase pattern of 512×512 for 4×4 splitting.

Fig. 3.
Fig. 3.

Steering grating: (a) diagram of quantization phase and slope, (b) the different mean slopes corresponding to the different group of phase values, (c) phase pattern of one-dimensional steering grating, which has 512×512 pixels.

Fig. 4.
Fig. 4.

Results of experiments: (a) the result of Dammann grating, (b) the result of the optimal splitting grating.

Fig. 5.
Fig. 5.

Results of experiments: (a) diffraction efficiency of two splitting gratings in different periods, (b) uniformity of two splitting gratings in different periods.

Fig. 6.
Fig. 6.

Combined phase patterns in experiments: (a) phase pattern by superimposing the Dammann grating with Airy cubic phase mask, (b) phase pattern by superimposing the optimal splitting grating with Airy cubic phase mask.

Fig. 7.
Fig. 7.

Results of experiments: (a) an array of 4×4 Airy beams by superimposing Dammann grating and cubic phase pattern, (b) an array of 4×4 Airy beams by superimposing optimal splitting grating and cubic phase pattern.

Fig. 8.
Fig. 8.

Intensity distribution of 2D Airy beam array at different distances from the focus of lens. (a)–(f) Airy beams array splitting by the Dammann grating at z=0mm, 2 mm, 4 mm, 6 mm, 8 mm, and 10 mm, respectively. (g)–(l) Airy beams array splitting by the optimal splitting grating at z=0mm, 2 mm, 4 mm, 6 mm, 8 mm, and 10 mm, respectively.

Fig. 9.
Fig. 9.

Experimental results: (a) transverse acceleration of the Airy beams array of Dammann grating, (b) transverse acceleration of the Airy beams array of optimal splitting grating.

Fig. 10.
Fig. 10.

Results of experiments: transverse acceleration of the Airy beam.

Fig. 11.
Fig. 11.

Results of experiments: (a) an array of 4×4 Airy beams scanning by superimposing Dammann grating, deflection grating, and cubic phase pattern; (b) an array of 4×4 Airy beams scanning by superimposing optimal splitting grating, deflection grating, and cubic phase pattern.

Fig. 12.
Fig. 12.

Results of experiments: deflection of the Airy beams array in y axis.

Fig. 13.
Fig. 13.

Experimental results: (a) the realized deflection angles and the theoretical deflection angles, (b) the error distribution of deflection angle.

Fig. 14.
Fig. 14.

Optical field distributions of a array of 1×4 Airy beams corresponding to different phase modulation depths.

Fig. 15.
Fig. 15.

Beams size correspond to the different phase modulation depth.

Tables (3)

Tables Icon

Table 1. Diffraction Efficiency of Different Period Dammann Grating

Tables Icon

Table 2. Diffraction Efficiency of Different Period Optimal Splitting Grating

Tables Icon

Table 3. Deflection of the Airy Beams in Transverse Acceleration Direction at Different Propagation Distances

Equations (15)

Equations on this page are rendered with MathJax. Learn more.

tDG(x)=m=cmexp(i2πmx),
cm={2n=0N(xn+1xn),m=01mπn=0N(1)n[sin(2πmxn+1)sin(2πmxn)],m0.
ak=12πππexp(iφOG(x))exp(ikx)dx,
exp(iφOG(x))=k=mmμkexp(iαk)exp(ikx)|k=mmμkexp(iαk)exp(ikx)|,
η=k=mm|ak|2k=|ak|2.
φSGideal(x,θ)=2πλxsinθ+φ0,
Δφideal(θ)=2πdλsinθ.
φSG(θ,j)=round(φSGideal(θ,j)M2π)2πM.
ϕ(x,z=0)=Ai(x/x0)exp(αx/x0),
Ai(x)=12πexp(iu3/3+ixu)du.
ϕ1(x,z=0)=Ai(x/x0)exp(ax/x0)·exp(iφDG(x)),
ϕ2(x,z=0)=Ai(x/x0)exp(ax/x0)·exp(iφOG(x)).
ϕ3(x,z=0)=Ai(x/x0)exp(ax/x0)·exp(iφDG(x))·exp(iφSG(x)),
ϕ4(x,z=0)=Ai(x/x0)exp(ax/x0)·exp(iφOG(x))·exp(iφSG(x)).
ε=(θrealizedθ)θspot×100%,

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